464 BELL SYSTEM TECHNICAL JOURNAL 



The power density spectrum of the errors is, from (2.1), 

 12/ = 4 / ^r cos lirfr dr 



= ^0 [ G{a) cos ItJt dr (2.12) 



Jo 



If the signal band is flat from / = to / = /o, with no energy outside this 

 band, 



a = - \ cos l-KTjdf = / — (2.13) 



Letting y = ///o, 



a(7) = ^^^ = -' f " G ('^) COS T. &, (2.14) 



To complete the calculation, we must evaluate the integral (2.10). The 

 first step is to transform the double summation (2.11) into products of single 

 sums by the change of indices: 



m' + n'\ 



m = 



m -{- n = m\ / 2 



or I (2.15) 



m — n = n' / \ m—n 



n = 



The rearrangement is permissible because the double series is absolutely 

 convergent. The new indices m' and n' also run from minus to plus infinity, 

 but must be either both even or both odd because m! ± n' is even. On 

 dropping the primes after the substitution is completed, we find 



Tj( \ v^ —k[2m{x + y) + Wl v" 

 H{x, y)= 2L> exp /\ 2^ 



w»=— 00 ~V-»- \^ **/ n=— 00 



4(1 + a) 

 • exp 



k[2n(x - y) + 4n'\ ^ ^ 



' exp 

 • exp 



4(1 - a) 



-mm + 1)(:^ + y) + (2m + l)'^ 

 4(1 + a) 



-k[2n + l)(x - y) + Cln + 1)'] 

 4(1 - a) 



(2.16) 



A further simplification results from a change of the variables of integration 

 to eliminate the terms in xy. This is done by setting 



/.= « + A ^ /.= (.+.)/2\ ^^^^^ 



\y — u — v/ \v = {x—y)/2/ 



